and we want to find all the submatrices (we just need the row indices and column indices of the corners) with these properties:

contain at least L ones and L zeros

contain max H elements

i.e. take the previous matrix with L=1 and H=5, the submatrix 1 2 1 4 (row indices 1 2 and column indices 1 4)

0 1 1 1
1 1 1 1

satisfies the property 1 but has 8 elements (bigger than 5) so it is not good;

the matrix 4 5 1 2

0 1
0 0

is good because satisfies both the properties.

The objective is then to find all the submatrices with min area 2*L, max area H and containg at least L ones and L zeros.

If we consider a matrix as a rectangle it is easy to find all the possibile subrectangles with max area H and min area 2*L by looking at the divisors of all the numbers from H to 2*L.

For example, with H=5 and L=1 all the possibile subrectangles/submatrices are given by the divisors of

H=5 -> divisors [1 5] -> possibile rectangles of area 5 are 1x5 and 5x1

4 -> divisors [1 2 4] -> possibile rectangles of area 4 are 1x4 4x1 and 2x2

3 -> divisors [1 3] -> possibile rectangles of area 3 are 3x1 and 1x3

2*L=2 -> divisors [1 2] -> possibile rectangles of area 2 are 2x1 and 1x2

I wrote this code, which, for each number finds its divisors and cycles over them to find the submatrices. To find the submatrices it does this: take for example a 1x5 submatrix, what the code does is to fix the first line of the matrix and move step by step (along all the columns of the matrix) the submatrix from the left edge of the matrix to the right edge of the matrix, then the code fixes the second row of the matrix and moves the submatrix along all the columns from left to right, and so on until it arrives at the last row.

It does this for all the 1x5 submatrices, then it considers the 5x1 submatrices, then the 1x4, then the 4x1, then the 2x2, etc.

The code do the job in 2 seconds (it finds all the submatrices) but for big matrices, i.e. 200x200, a lot of minutes are needed to find all the submatrices. So I wonder if there are more efficient ways to do the job, and eventually which is the most efficient.

This is my code:

clc;clear all;close all
%% INPUT
P= [0 1 1 1 0 0 0 ;
1 1 1 1 0 1 1 ;
0 0 1 0 0 1 0 ;
0 1 1 0 1 1 1 ;
0 0 0 0 0 0 1 ;
0 0 0 0 0 0 1];
L=1; % a submatrix has to containg at least L ones and L zeros
H=5; % max area of a submatrix
[R,C]=size(P); % rows and columns of P
sub=zeros(1,6); % initializing the matrix containing the indexes of each submatrix (columns 1-4), their area (5) and the counter (6)
counter=1; % no. of submatrices found
%% FIND ALL RECTANGLES OF AREA >= 2*L & <= H
%
% idea: all rectangles of a certain area can be found using the area's divisors
% e.g. divisors(6)=[1 2 3 6] -> rectangles: 1x6 6x1 2x3 and 3x2
tic
for sH = H:-1:2*L % find rectangles of area H, H-1, ..., 2*L
div_sH=divisors(sH); % find all divisors of sH
disp(['_______AREA ', num2str(sH), '_______'])
for i = 1:round(length(div_sH)/2) % cycle over all couples of divisors
div_small=div_sH(i);
div_big=div_sH(end-i+1);
if div_small <= R && div_big <= C % rectangle with long side <= C and short side <= R
for j = 1:R-div_small+1 % cycle over all possible rows
for k = 1:C-div_big+1 % cycle over all possible columns
no_of_ones=length(find(P(j:j-1+div_small,k:k-1+div_big))); % no. of ones in the current submatrix
if no_of_ones >= L && no_of_ones <= sH-L % if the submatrix contains at least L ones AND L zeros
% row indexes columns indexes area position
sub(counter,:)=[j,j-1+div_small , k,k-1+div_big , div_small*div_big , counter]; % save the submatrix
counter=counter+1;
end
end
end
disp([' [', num2str(div_small), 'x', num2str(div_big), '] submatrices: ', num2str(size(sub,1))])
end
if div_small~=div_big % if the submatrix is a square, skip this part (otherwise there will be duplicates in sub)
if div_small <= C && div_big <= R % rectangle with long side <= R and short side <= C
for j = 1:C-div_small+1 % cycle over all possible columns
for k = 1:R-div_big+1 % cycle over all possible rows
no_of_ones=length(find(P(k:k-1+div_big,j:j-1+div_small)));
if no_of_ones >= L && no_of_ones <= sH-L
sub(counter,:)=[k,k-1+div_big,j,j-1+div_small , div_big*div_small, counter];
counter=counter+1;
end
end
end
disp([' [', num2str(div_big), 'x', num2str(div_small), '] submatrices: ', num2str(size(sub,1))])
end
end
end
end
fprintf('\ntime: %2.2fs\n\n',toc)